A Simple yet Exact Analysis of the MultiQueue

TL;DR

This paper provides a simpler, exact analysis of the expected rank error in the MultiQueue, a relaxed concurrent priority queue, revealing it to be approximately 5/6 of the number of internal queues, and extends the analysis to generalized deletion schemes.

Contribution

It introduces a straightforward method to exactly determine the long-term rank error of the MultiQueue, improving upon previous complex potential-based analyses.

Findings

Expected rank error is approximately 5/6 of n.

Analysis applies to generalized deletion schemes based on queue rank.

Long-term distribution of rank error is derived exactly.

Abstract

The MultiQueue is a relaxed concurrent priority queue consisting of $n$ internal priority queues, where an insertion uses a random queue and a deletion considers two random queues and deletes the minimum from the one with the smaller minimum. The rank error of the deletion is the number of smaller elements in the MultiQueue. Alistarh et al. [2] have demonstrated in a sophisticated potential argument that the expected rank error remains bounded by $O(n)$ over long sequences of deletions. In this paper we present a simpler analysis by identifying the stable distribution of an underlying Markov chain and with it the long-term distribution of the rank error exactly. Simple calculations then reveal the expected long-term rank error to be $\tfrac{5}{6}n-1+\tfrac{1}{6n}$. Our arguments generalize to deletion schemes where the probability to delete from a given queue depends only on the rank of the queue. Specifically, this includes deleting from the best of $c$ randomly selected queues for any $c>1$.